Defined below are all of the functions we will use in our analysis. They are all inspired or directly taken from Justin Bois.

We then extract the catastrophe times for only the experiments with 12 uM of tubulin to compare our two models.

\begin{align} Gamma\ Distribution\ PDF\ f(t;\alpha, \beta) = \frac{1}{\Gamma(\alpha)} \frac{(\beta t)^\alpha}{t} \mathrm{e}^{-\beta t} \end{align}\begin{align} Mixture\ Model\ PDF\ f(t;\beta_1, \beta_2) = \frac{\beta_1\beta_2}{\beta_2 - \beta_1}\left(\mathrm{e}^{-\beta_1 t} - \mathrm{e}^{-\beta_2 t}\right). \end{align}

Gamma

We first obtain the MLEs for α and β parameters of the gamma model. We also compute the 95% confidence interval for the parameters.

Two Successive Poisson Processes Mixture Model

Then, we obtain the MLEs for both βs in the mixture model with 95% confidence intervals.

We then used the MLEs for both sets of parameters to generate predictive ecdfs to compare the models to the actual data and against each other.

From the plots above, while the mixture model does not appear to be modeling the actual phenomenon "perfectly", it is certainly doing a better job than the gamma model. From this graphical model assement, we decided to move forward with the mixture model over the gamma model. We then compute the MLEs for each concentration of tubulin. From the MLEs, it seems that the experiments with lower amounts of tubulin had shorter lengths of time between the catastrophes.